### "The classical world is not created by observation... but rather by the lack of it" (N. P. Landsman)

**Quantum physics (construction = classical mechanics re)construction**

The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantum-theoretical... reinterpretation... of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand,no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail.

On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely in the limit h→0 of small Planck’s constant (in a finite system), in the limit N→∞ of a large system with N degrees of freedom (at fixed h), and through decoherence and consistent histories.The first limit is closely related to modern quantization theory and microlocal analysis, whereas the second involves methods of C^{∗}-algebras and the concepts of superselection sectors and macroscopic observables.In these limits, the classical world does not emerge as a sharply defined objective reality, but rather as an approximate appearance relative to certain “classical” states and observables. Decoherence subsequently clarifies the role of such states, in that they are “einselected”, i.e. robust against coupling to the environment. Furthermore, the nature of classical observables is elucidated by the fact that they typically define (approximately) consistent sets of histories. This combination of ideas and techniques does not quite resolve the measurement problem, but it does make the point that classicality results from the elimination of certain states and observables from quantum theory. Thus the classical world is not created by observation (as Heisenberg once claimed), but rather by the lack of it...

We will discuss these ideas in more detail below, and indeed our discussion of the relationship between classical and quantum mechanics will be partly historical. However, other than that it will be technical and mathematically rigorous.For the problem at hand is so delicate that in this area sloppy mathematics is almost guaranteed to lead to unreliable physics and conceptual confusion(notwithstanding the undeniable success of poor man’s math elsewhere in theoretical physics).Except for von Neumann, this was not the attitude of the pioneers of quantum mechanics; but while it has to be acknowledged that many of their ideas are still central to the current discussion, these ideas per se have not solved the problem...

For even if it is granted that decoherence yields the disappearance of superpositions of Schrödinger cat type, or that consistent historians give us consistent sets none of whose elements contain such superpositions among their properties, this by no means suffices to explain the emergence of classical phase spaces and flows thereon determined by classical equations of motion.Since so far the approaches cited in Sections 5 and 6 have hardly been combined with the decoherence and/or the consistent histories program,a full explanation of the classical world from quantum theory is still in its infancy. This is not merely true at the technical level, but also conceptually; what has been done so far only represents a modest beginning. On the positive side, here lies an attractive challenge for mathematically minded researchers in the foundations of physics!

(Submitted on 10 Jun 2005 (v1), last revised 25 Jul 2005 (this version, v2))

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